### Abstract

Let (X_{1}, . . . , X_{n}) be independently and identically distributed observations from an exponential family p_{θ} equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ_{0}, admits an asymptotic expansion in terms of the Fisher information I(θ_{0}), the prior w, and their first two derivatives. The leading term of the expansion is of the form n^{d/2}C_{1}(d, θ_{0}) and the second order term is of the form n^{d/2-1}c_{2}(d, θ_{0}, w), with an error term that is o(n^{d/2-1}). We identify the functions C_{1} and C_{2} explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ_{0}. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Original language | English (US) |
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Pages (from-to) | 163-185 |

Number of pages | 23 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1999 |

### Keywords

- Asymptotics
- Bayes factor
- Chi-squared distance
- Expected posterior
- Relative entropy

### ASJC Scopus subject areas

- Statistics and Probability

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## Cite this

*Annals of the Institute of Statistical Mathematics*,

*51*(1), 163-185. https://doi.org/10.1023/A:1003891404142